On stable sl3-homology of torus knots - arXiv

ON STABLE sl 3 -HOMOLOGY OF TORUS KNOTS

arXiv:1404.0623v1 [math.GT] 2 Apr 2014

EUGENE GORSKY AND LUKAS LEWARK

A BSTRACT. The stable Khovanov-Rozansky homology of torus knots has been conjecturally described as the Koszul homology of an explicit non-regular sequence of polynomials. We verify this conjecture against newly available computational data for sl 3 -homology. Special attention is paid to torsion. In addition, explicit conjectural formulae are given for the sl N -homology of (3, m)-torus knots for all N and m, which are motivated by higher categorified Jones-Wenzl projectors. Structurally similar formulae are proven for Heegard-Floer homology.

1. I NTRODUCTION Torus knots are among the simplest and best understood knots, and their invariants have been studied and computed using various mathematical and physical tools. For example, Jones and Rosso [12] computed all sl N -quantum invariants of torus knots explicitly. The computation of Khovanov and Khovanov-Rozansky homology of torus knots remains an important open problem in knot theory. The experimental computations of Bar-Natan, Shumakovitch, Turner [2, 3, 19, 20] and others revealed a rich structure, including nontrivial torsion, in sl 2 -homology of torus knots. On the other hand, the main conjecture of [11] related the HOMFLY-PT homology of (n, m)-torus knots to finite dimensional representations of rational Cherednik algebras with parameter m/n. In the limit m → ∞, this construction has been used in [10] for a conjectural description of stable sl N -homology of torus knots. For N = 2, this conjecture has been verified against the experimental data for stable torus knots with up to 8 strands, and they agree both in reduced and in unreduced homology. Conjecture 1. ([10]) Consider the triply graded algebra Hn with even generators x0 , . . . , xn−1 and odd generators ξ0 , . . . , ξn−1 of degrees deg xk = q 2k+2 t2k ,

deg ξk = a2 q 2k t2k+1 .

It is equipped with the family of Koszul differentials dN defined by the equations: X (1) dN (xk ) = 0, dN (ξk ) = xi 1 · . . . · xi N . i1 +...+iN =k

Then the stable sl N -Khovanov-Rozansky homology of the (n, ∞)-torus knot is isomorphic to H ∗ (Hn , dN ) after the regrading a = q N . Recently the second named author developed a computer program [13, 14] that allows one to compute integral sl 3 -Khovanov-Rozansky homology of any knot, in reasonable time for knots such as the (5, 14)- or (6, 7)-torus knots. The main goal of this article is to verify Conjecture 1 for N = 3 using the newly available experimental data, see Section 4. In Section 2, we study Conjecture 1, proving it to be correct for the (2, ∞)-torus knot, and giving an explicit formula for the 3-strand case. Reduced homology is related to unreduced. We Date: April 2, 2014. The first author is partially supported by the grants RFBR13-01-00755, NSh-4850.2012.1. The second author is supported by the EPSRC-grant EP/K00591X/1. 1

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show that for a prime p > N + 1, the Koszul model for stable sl N -homology of the p-strand torus knot has p-torsion, and give a formula for the Koszul model of sl N -homology with ZN coefficients, which has a relatively simple form. Note that while a definition of sl N -homology with integer or Zp -coefficients may be expected to exist, it has so far only been given for N = 2 and N = 3. For 3-strand finite torus knots T (3, m), we also write down explicit conjectural formulae for the Poincaré polynomials of sl N -homology (see Section 3). Namely, for each of the 4 standard Young tableaux T with 3 boxes we define a certain explicit rational function PT (N ). Conjecture 2. The Poincaré series of the unreduced sl N -homology of the (3, 3k + 1)-torus knot equals (up to an overall shift of q 3k(N −1)−2 ): P(T (3, 3k + 1), sl N ) = P 1

2 3

(N ) + q 6k t4k P 1

2

(N ) + q 6k t4k P 1

3

3

(N ) + q 12k t6k P 1 (N ).

2

2 3

The Poincaré series of the unreduced sl N -homology of the (3, 3k + 2)-torus knot equals (up to an overall shift of q 3k(N −1)−3 ): P(T (3, 3k + 2), sl N ) = P 1

2 3

(N ) + q 6k t4k P 1

2

(N )+

3

q 6k+4 t4k+2 P 1

3

(N ) + q 12k+4 t6k+2 P 1 (N ).

2

2 3

The conjecture is motivated by the work [9], where the “refined Chern-Simons invariants” [1] of (n, m)-torus knots were expressed as sums over standard Young tableaux of size n. It was conjectured in [1] that these invariants agree with the Poincaré polynomials of HOMFLY-PT homology. We show that for n = 3 (and n = 2) the HOMFLY-PT analogues of PT (N ) can be viewed as Poincaré series of certain free supercommutative algebras, and PT (N ) can be obtained as homology of certain differentials dN acting on these algebras. In particular, P 1 2 3 (N ) coincides with the Poincaré series of stable sl N -homology of T (3, ∞) which, by Conjecture 1, can be described as homology of dN . This decomposition is also expected to be related to the categorification of higher Jones-Wenzl projectors [5], but we do not pursue this relation here. Surprisingly enough, we prove that the Poincaré polynomials of the Heegaard-Floer homology of T (3, m) can be written using a formula with a similar structure. The corresponding differential corresponds to d0 in the notations of [7, 11] and to d1|1 in the notations of [8]. ACKNOWLEDGMENTS The authors would like to thank M. Khovanov, A. Lobb and J. Rasmussen for the useful discussions and remarks. 2. T HE KOSZUL MODEL FOR STABLE sl N - HOMOLOGY 2.1. Koszul model. In [10, 11] the following algebraic model for the stable homology of torus knots was proposed. Conjecture 1. ([10]) Consider the triply graded algebra Hn with even generators x0 , . . . , xn−1 and odd generators ξ0 , . . . , ξn−1 of degrees deg xk = q 2k+2 t2k ,

deg ξk = a2 q 2k t2k+1 .


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It is equipped with the family of Koszul differentials dN defined by the equations: X (1) dN (xk ) = 0, dN (ξk ) = xi 1 · . . . · xi N . i1 +...+iN =k

Then the stable sl N -Khovanov-Rozansky homology of the (n, ∞)-torus knot is isomorphic to H ∗ (Hn , dN ) after the regrading a = q N . Example 3. For n = 1 one has a single even generator x0 and a single odd generator ξ0 , and the differential has the form dN (ξ0 ) = xN 0 . The homology has dimension N and is spanned by N −1 1, x0 , . . . , x0 , so it is indeed isomorphic to the sl N -homology of the unknot. We denote the homology of dN as Hnalg (sl N ). After setting a = q N the variables are graded as follows: deg xk = q 2k+2 t2k , deg ξk = q 2N +2k t2k+1 , so that dN preserves the q-grading and decreases the t-grading by 1. P P∞ k k 2.2. Generators. Consider the generating functions x(τ ) = ∞ k=0 xk τ and ξ(τ ) = k=0 ξk τ . The equation (1) can be rewritten as (2)

dN (x(τ )) = 0,

dN (ξ(τ )) = x(τ )N

mod τ n .

Remark that (3)

˙ )) = N x(τ dN (ξ(τ ˙ )x(τ )N −1

Define µ(τ ) =

∞ X

mod τ n .

˙ ), µk τ k−1 := N x(τ ˙ )ξ(τ ) − x(τ )ξ(τ

k=1 P so µk = i+j=k (N i − j)xi ξj . It follows from (2) and (3) that dN (µ(τ )) = 0, so dN (µk ) = 0 for all k. The degree of µk equals q 2N +2k+2 t2k+1 .

Conjecture 4. The homology of dN is generated as an algebra by xk and µk . 2.3. Reduced homology. The reduced homology is defined as a quotient of the unreduced homology by the homology of the unknot, i.e. by the equation x0 = 0. It turns out that the reduced and unreduced Koszul homology are tightly related. Theorem 5. We have the following isomorphism: alg Hnalg,red (sl N ) ' Z[ξ1 , . . . , ξN −1 , xn−N +1 , . . . , xn ] ⊗ Hn−N (sl N ).

Proof. In reduced homology we set x0 = 0, so dN (ξi ) = 0 for i < N . Furthermore, dred N (ξi ) red N coincides with dN (ξi−N ) where all xi are replaced by xi−1 . For example, dN (ξN ) = x1 . Note that the isomorphism does not preserve q- and t-gradings, but it is easy to reconstruct the change of gradings transforming xi to xi+1 and ξi to ξi+N . 2.4. ZN -homology for prime N . In this subsection we give a simple description of sl N homology with ZN -coefficients, assuming for simplicity that N is a prime number. Theorem 6. Suppose that N is prime. Then the algebraic Poincaré series has the form: n−1

b N c 1 + q 2N +2k t2k+1 Y 1 − q 2N +2iN t2iN alg Pn (sl N , ZN ) = . 2k+2 t2k 2N +2iN t2iN +1 1 − q 1 + q i=0 k=0 n−1 Y

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Proof. If N is prime, then dN (ξ(τ )) = x(τ )N ≡ Therefore

X

iN xN i τ

( 0 mod N, dN (ξk ) ≡ xN mod N, i

mod N.

k= 6 Ni k = N i.

The homology is generated by xi modulo relations xN i = 0 and by ξk for k not divisible by N. See Table 1 for an illustration of the theorem. 2.5. Torsion. The computations of Khovanov homology showed the presence of large and complicated torsion. The first computations in sl 3 -knot homology indeed confirmed the existence of torsion as well. It has been shown in [10] that the algebraic model is compatible at least with some part of this torsion, the following result generalizes this check to sl N -homology. Theorem 7. Let p > N + 1 be a prime number, then the algebraic homology Hpalg (sl N , Z) has p-torsion in bidegree q 2p+2N +2 t2p+1 . P Proof. Recall that for µp = pi=0 (N i − p + i)xi ξp−i one has dN (µp ) = 0. This generator is not present in Hp , since xp and ξp are not present in the Koszul complex. Consider the element tp :=

p−1 X

(N i − p + i)xi ξp−i = µp − N pxp ξ0 + px0 ξp ,

i=1

which is present in Hp . We have dN (tp ) = p(x0 dN (ξp ) − N xp xN 0 ), so dN (tp ) vanishes modulo p. Since tp has a term (N − p + 1)x1 ξp−1 that does not vanish modulo p, the dimension of the kernel of dN over Zp is bigger than its rank over Z. Example 8. For N = 3 and p = 5 the computations of [14] show Z5 -torsion in bidegree q 18 t11 of the homology of the (5, m)-torus knot for 6 ≤ m ≤ 14. It is plausible that it survives in the stable limit and hence agrees with the torsion generator t5 . See Table 2 for the case m = 9. Indeed, the torsion classes described in Theorem 7 (and their products with x’s and µ’s) do not cover all stable torsion. Example 9. Consider the following class in Hn (defined for n ≥ 5): A := 4x0 ξ1 ξ4 − 8x1 ξ0 ξ4 + 8x4 ξ0 ξ1 − 2x0 ξ2 ξ3 − 2x2 ξ1 ξ2 − 4x3 ξ0 ξ2 + x1 ξ1 ξ3 + 4x2 ξ0 ξ3 . One can check that d2 (A) = 10x1 x3 (2x1 ξ0 − x0 ξ1 ), so A represents a class in Hnalg (sl 2 , Z5 ). On the other hand, one can check (say, from Conjecture 4) that A cannot be lifted to a class in Hnalg (sl 2 , Z), hence Hnalg (sl 2 , Z) has 5-torsion in bidegree q 20 t12 for all n ≥ 5. This torsion can be seen in the data of [3], e.g. for the (5, 6)-, (6, 7)-, (7, 8)- and (8, 9)- torus knots. Example 10. Consider the following class in H6 : B := 5x21 ξ5 −5x1 x5 ξ1 −10x1 x4 ξ2 +16x2 x4 ξ1 −15x1 x3 ξ3 +15x23 ξ1 +3x22 ξ3 −3x2 x3 ξ2 −6x1 x2 ξ4 . One can check that d3 (B) = −105x0 x21 x2 x3 , so B represents a class in H6alg (sl 3 , Z7 ). Moreover, B cannot be lifted to a class in H6alg (sl 3 , Z), so H6alg (sl 3 , Z) has 7-torsion in bidegree q 24 t15 . This is also confirmed by the experimental data [14], i.e. by the computed homology of the

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TABLE 1. The unreduced sl 3 -homology of T (5, 9) with Z3 -coefficients, cf. Theorem 6. The top row indicates the homological degree t, and the left-most column ‡ = q − 2t, where q is the quantum degree. For simplicity, we follow the grading convention (which is different from the Khovanov and Rozansky’s original one) that leads to e.g. the reduced sl 3 -homology of the trefoil with positive crossings having Poincaré polynomial q 4 + t2 q 8 + t3 q 12 . Note also that the stable part of an actual (n, m)-torus knot is shifted in the quantum degree by 2(nm − n − m) relative to the homology of the (n, ∞)-torus knot. Each cell shows the dimension of homology at the corresponding degree. The first divergence from the stable model (at homological degree 16) is indicated by a doubled vertical line.

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ON STABLE sl 3 -HOMOLOGY OF TORUS KNOTS 5

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TABLE 2. The unreduced sl 3 -homology of T (5, 9). See the caption of Table 1 for detailed explanations. The Z5 -torsion is printed at the lower left corner of each cell in bold and green; for the sake of legibility, Z3 -torsion is not printed, but cf. Table 1. The only other torsion is Z2 -torsion (t23 q 36 + t24 q 42 ), which is not printed either, and may be expected to be unstable. Note the presence of Z5 -torsion at t11 q 18 (indicated by a box), in accordance with Theorem 7.

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6 EUGENE GORSKY AND LUKAS LEWARK


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(6, 7)-torus knot and (6, 8)-torus link. Note that B ≡ x2 µ5 mod 5, so it does not contribute to the 5-torsion. Remark 11. Przytycki and Sazdanović conjectured in [16, Conjecture 6.1] that the closure of an n-strand braid cannot have Zp -torsion in Khovanov homology for a prime p > n. Example 10 shows that a naive generalization of this conjecture (“the closure of an n-strand braid cannot have Zp -torsion in sl N -homology for a prime p > max(n, N )”) does not hold for sl 3 -homology, since the (6, 7)-torus knot and the (6, 8)-torus link already have 7-torsion. 2.6. (2, ∞) case. For n = 2 we have two even generators x0 , x1 and two odd generators ξ0 , ξ1 , with the differential −1 dN (ξ0 ) = xN dN (ξ1 ) = N xN x1 . 0 , 0 2N +4 3 There is a generator µ1 = N x1 ξ0 − x0 ξ1 of degree q t , the homology is generated by x0 , x1 and µ1 modulo relations xN 0 = 0,

−1 N xN x1 = 0, 0

−1 xN µ1 = 0. 0

The monomial basis in Q-homology has the form −1 xN , xi0 xa1 µb1 , i < N − 1, b ≤ 1, 0

so the Poincaré series has the form (4)

P2 (q, t) = q 2N −2 +

(1 − q 2N −2 )(1 + q 2N +4 t3 ) (1 − q 2 )(1 − q 4 t2 )

1 − q 2N − q 2N +2 t2 + q 2N +4 t2 + q 2N +4 t3 − q 4N +2 t3 . (1 − q 2 )(1 − q 4 t2 ) The actual sl N -Khovanov-Rozansky homology was computed for all N by Cautis in [4, Corollary 10.2]. Since the two answers coincide, we obtain the following result. =

Theorem 12. Conjecture 1 is true for n = 2 and all N . 2.7. (3, ∞) case. For n = 3 we have new generators x2 and ξ2 in H3 , and the differential looks like N N −2 2 N −1 dN (ξ2 ) = N x0 x2 + x x1 . 2 0 We have a new homology generator µ2 = 2N x2 ξ0 + (N − 1)x1 ξ1 − 2x0 ξ2 ,

deg µ2 = q 2N +6 t3

and one can explicitly check the identity dN (µ2 ) = 0. In zeroth Koszul homology we get N N −2 2 N −1 N N −1 x x1 . Q[x0 , x1 , x2 ]/ x0 , N x0 x1 , N x0 x2 + 2 0 −2 3 If we multiply the third term by x1 and subtract x2 times the second one, we get xN x1 = 0. 0 N −1 N −1 −2 3 N The monomial basis consists of all monomials not divisible by x0 , x0 x1 , x0 x2 and xN x1 , 0 and the Poincaré series equals

P3a=0 (q, t) = q 2N −2 + =

1 − q 2N −2 q 2N +8 t6 − (1 − q 2 )(1 − q 4 t2 )(1 − q 6 t4 ) (1 − q 4 t2 )(1 − q 8 t6 )

1 − q 2N − q 2N +2 t2 − q 2N +4 t4 + q 2N +4 t2 + q 2N +6 t4 . (1 − q 2 )(1 − q 4 t2 )(1 − q 6 t4 )

8


Let us describe the relations between µ’s. We have −1 dN (ξ0 ξ1 ) = xN µ1 , 0

N −1 −2 2 dN (2ξ0 ξ2 ) = 2xN x2 + N (N − 1)xN x1 ξ0 0 ξ2 − 2N x0 0 −2 = −x0N −1 µ2 − (N − 1)xN x1 µ 1 , 0

−1 −2 2 dN (2ξ1 ξ2 ) = 2N x0N −1 x1 ξ2 − 2N xN x2 + N (N − 1)xN x1 ξ1 0 0 −2 −2 = −N xN x1 µ2 + 2N xN x2 µ 1 . 0 0 −1 −1 −2 Therefore the monomials cannot be divisible by xN µ 1 , xN µ2 and xN x1 µ2 , and the 0 0 0 Poincaré series equals

P3a=1 (q, t)

(1 − q 2N −2 )(q 2N +4 t3 + q 2N +6 t5 ) − q 4N +6 t7 (1 − q 4 ) . = (1 − q 2 )(1 − q 4 t2 )(1 − q 6 t4 )

Finally, the homology at level 2 is generated by µ1 µ2 and the unique relation has the form: −2 dN (2ξ0 ξ1 ξ2 ) = xN µ1 µ2 , 0

hence P3a=2 (q, t)

q 4N +10 t8 (1 − q 2N −4 ) = . (1 − q 2 )(1 − q 4 t2 )(1 − q 6 t4 )

We obtain the following result: Theorem 13. The Poincaré series for the rational homology H3alg (sl N , Q) is given by the formula P3 (sl N , Q)(1 − q 2 )(1 − q 4 t2 )(1 − q 6 t4 ) = 1 − q 2N − q 2N +2 t2 + q 2N +4 (t2 + t3 − t4 ) + q 2N +6 (t4 + t5 ) − q 4N +2 t3 − q 4N +4 t5 − q 4N +6 t7 + q 4N +10 (t7 + t8 ) − q 6N +6 t8 . 3. P ROJECTORS AND sl N - HOMOLOGY OF (3, m)- TORUS KNOTS In this section we give a conjectural formula for sl N -homology of (3, m)-torus knots for all N and m. The construction is motivated by the study of higher categorified Jones-Wenzl projectors, introduced by Cooper and Hogancamp [5], although we do not pursue this analogy here. These projectors are labeled by the standard Young tableaux (SYT) and categorify the projectors onto irreducible summands in the regular representation of the Hecke algebra. By a theorem of Rozansky [18] (see also [4, 17]), the homology of the categorified projector corresponding to the symmetric representation of Hn coincides with the stable homology of the (n, ∞)-torus knot. 3.1. Decomposition of the homology. We expect the Poincaré polynomials of homology of three-strand torus knots to be defined by the following conjecture. Conjecture 2. The Poincaré series of the unreduced sl N -homology of the (3, 3k + 1)-torus knot equals (up to an overall shift of q 3k(N −1)−2 ): P(T (3, 3k + 1), sl N ) = P 1

2 3

(N ) + q 6k t4k P 1 3

2

(N ) + q 6k t4k P 1 2

3

(N ) + q 12k t6k P 1 (N ). 2 3


9

The Poincaré series of the unreduced sl N -homology of the (3, 3k + 2)-torus knot equals (up to an overall shift of q 3k(N −1)−3 ): P(T (3, 3k + 2), sl N ) = P 1

(N ) + q 6k t4k P 1

2 3

2

(N )+

3

q 6k+4 t4k+2 P 1

3

(N ) + q 12k+4 t6k+2 P 1 (N ).

2

2 3

The conjecture is expected to hold both for reduced and unreduced sl N -and HOMFLY-PT homology of (3, m)-torus knots, and we prove below that a similar decomposition holds for the Heegaard-Floer homology, too. For N = 2, the answer agrees with the computations of Turner [20]. For N = 3, we checked the answer on the computer both for large (k = 28) and small torus knots and found a perfect agreement with the experimental data. The existence of such decompositions was first conjectured by Oblomkov and Rasmussen [15], but, to the knowledge of the authors, the explicit formulae for PT have not been written down before. 3.2. Symmetric projector, two strands. The unreduced HOMFLY-PT homology of the symmetric projector is a free algebra with generators x0 , x1 , ξ0 , ξ1 with the degrees: deg(x0 ) = q 2 ,

deg(x1 ) = q 4 t2 ,

deg(ξ0 ) = a2 t,

deg(ξ1 ) = a2 q 2 t3 .

Therefore the Poincaré series of this homology equals: (1 + a2 t)(1 + a2 q 2 t3 ) . 2 (1 − q 2 )(1 − q 4 t2 ) The reduced HOMFLY-PT homology is generated by x1 and ξ1 , and the Poincaré series equals: P1

=

(1 + a2 q 2 t3 ) . (1 − q 4 t2 ) According to Conjecture 1, the sl N -homology can be described as the homology of the differential dN after regrading a = q N . The unreduced sl N -homology of the symmetric projector are then given by (4): P red = 1 2

P d1N 2 =

1 − q 2N − q 2N +2 t2 + q 2N +4 t2 + q 2N +4 t3 − q 4N +2 t3 . (1 − q 2 )(1 − q 4 t2 )

Remark that the Euler characteristic of this projector is equal to the S 2 -colored sl N -invariant of the unknot. In the reduced homology one has dN = 0, so P d1N ,red = 2

(1 + q 2N +2 t3 ) . (1 − q 4 t2 )

3.3. Antisymmetric projector, two strands. The homology of the antisymmetric projector can be computed as follows. The HOMFLY-PT homology is an algebra with two even generators x0 , a1 and two odd generators ξ0 , θ1 . The degrees are equal to: deg x0 = q 2 ,

deg a1 = q −4 t−2 ,

deg ξ0 = a2 t,

Therefore the Poincaré series of this homology equals: P1 = 2

(1 + a2 t)(1 + a2 q −2 t) . (1 − q 2 )(1 − q −4 t−2 )

deg θ1 = q −2 a2 t.

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TABLE 3. The unreduced rational sl 3 -homology of T (3, 16). See the caption of Table 1 for detailed explanations. The first half of the table depicts the stable part, and the second half the part diverging from T (3, ∞).

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2

4

1

1

0

6

t

−2

‡

0

2

4

6

‡

−10

−8

−6

−4

−2

t

10 EUGENE GORSKY AND LUKAS LEWARK


11

The reduced Poincaré series equals P red = 1 2

(1 + a2 q −2 t) . (1 − q −4 t−2 )

The differential is given by the formula dN (ξ0 ) = xN 0 ,

−1 . dN (θ1 ) = N xN 0

−1 The homology is generated by x0 , a1 and N ξ0 − x0 θ1 modulo relation xN = 0. The Poincaré 0 series has the form: (1 − q 2N −2 )(1 + q 2N t) P d1N = . (1 − q 2 )(1 − q −4 t−2 )

2

The reduced Poincaré series equals: P d1N ,red = 2

(1 + q 2N −2 t) . (1 − q −4 t−2 )

There is a natural embedding Halg (1, ∞) ,→ Halg (2, ∞), and the Poincaré series of the quotient equals −P 1 . In other words, we have an identity 2

P1

2

+P1 =P1. 2

alg

alg Indeed, in H (1, ∞) we have generators ξ0 , x0 and the differential dN (ξ0 ) = xN 0 . In H (2, ∞) −1 we add the generators ξ1 , x1 and the differential dN (ξ1 ) = N xN x1 . The quotient is spanned 0 by the monomials containing either x1 or ξ1 . If we introduce the formal variable θ1 = ξ1 /x1 , then the quotient would be spanned by all monomials divisible by x1 . On the other hand, −1 dN (θ1 ) = N xN . 0

3.4. Homology of (2, m)-torus knots. The unreduced sl N -homology of (2, m)-torus knots was computed by Cautis in [4, Example 10.3.3], and can be reformulated as following: Theorem 14. ([4]) The Poincaré polynomial of the sl N -homology of the (2, 2k + 1)-torus knot is given by the following equation: P(T (2, 2k + 1), sl N ) = P d1N 2 + q 4k t2k P d1N . 2

The same decomposition holds for the reduced homology. The reduced homology of (2, 2k + 1)-torus knots is well known, see e.g. [7]. 3.5. Symmetric projector, three strands. As above, the HOMFLY-PT homology of the symmetric projector is expected to coincide with the algebra H3 , its Poincaré series equals (cf. [7]): (1 + a2 t)(1 + a2 q 2 t3 )(1 + a2 q 4 t5 ) (1 + a2 q 2 t3 )(1 + a2 q 4 t5 ) red P1 2 3 = , P = . 1 2 3 (1 − q 2 )(1 − q 4 t2 )(1 − q 6 t4 ) (1 − q 4 t2 )(1 − q 6 t4 ) The unreduced sl N -homology of the symmetric projector is given by Theorem 13: 1 P d1N 2 3 = (1 − q 2N − q 2N +2 t2 + q 2N +4 (t2 + t3 − t4 )+ 2 (1 − q )(1 − q 4 t2 )(1 − q 6 t4 ) q 2N +6 (t4 + t5 ) − q 4N +2 t3 − q 4N +4 t5 − q 4N +6 t7 + q 4N +10 (t7 + t8 ) − q 6N +6 t8 ).

12


In the reduced homology we have dN = 0 for N > 2 and d2 (ξ2 ) = x21 . Therefore P d1N ,red = 2 3

(1 + q 2N +2 t3 )(1 + q 2N +4 t5 ) , N > 2; (1 − q 4 t2 )(1 − q 6 t4 )

P d12 ,red = 2 3

(1 − q 8 t4 )(1 + q 6 t3 ) . (1 − q 4 t2 )(1 − q 6 t4 )

We also introduce a differential d0 on the reduced homology by the formula d0 (ξ2 ) = x1 . It is clear that (1 + a2 q 2 t3 ) = . P d10 ,red 2 3 (1 − q 6 t4 ) 3.6. Antisymmetric projector, three strands. The HOMFLY-PT homology of the antisymmetric projector is an algebra with two even generators x0 , a1 , a2 and two odd generators ξ0 , θ1 , θ2 . The degrees are equal to: deg x0 = q 2 ,

deg a1 = q −4 t−2 ,

deg a2 = q −6 t−2 ,

deg ξ0 = a2 t,

deg θ1 = a2 q −2 t,

deg θ2 = a2 q −4 t.

The Poincaré series equals: P1 = 2 3

(1 + a2 t)(1 + a2 q −2 t)(1 + a2 q −4 t) , (1 − q 2 )(1 − q −4 t−2 )(1 − q −6 t−2 )

P red = 1 2 3

(1 + a2 q −2 t)(1 + a2 q −4 t) . (1 − q −4 t−2 )(1 − q −6 t−2 )

The differential is given by the formula dN (ξ0 ) =

xN 0 ,

dN (θ1 ) =

N N −2 dN (θ2 ) = x . 2 0

−1 N xN , 0

The homology has even generators x0 , a1 , a2 and two odd generators N ξ0 − x0 θ1 , N2−1 θ1 − x0 θ2 modulo relation x0N −2 = 0, so P d1N = 2 3

(1 − q 2N −4 )(1 + q 2N −2 t)(1 + q 2N t) . (1 − q 2 )(1 − q −4 t−2 )(1 − q −6 t−2 )

Note that for N = 2 this homology vanishes, as expected. In reduced homology dN = 0, so dN ,red

P1 2 3

(1 + q 2N −2 t)(1 + q 2N t) . = (1 − q −4 t−2 )(1 − q −6 t−2 )

The differential d0 on reduced homology is given by the formula d0 (θ2 ) = a1 , and P d10 ,red = 2 3

(1 + a2 q −2 t) . (1 − q −6 t−2 )

3.7. Hook-shaped projectors. We formally introduce two power series P 1 3

that the following equations hold: P1

2

=P1

2 3

+P1 3

2

,

P1 =P1 +P1 2

2 3

2

3

.

2

and P 1 2

3

such


13

One can check that for HOMFLY-PT homology these series are given by P1

2

=

(1 + a2 t)(1 + a2 q −2 t)(1 + a2 q 2 t3 ) , (1 − q 2 )(1 − q 4 t2 )(1 − q −6 t−4 )

3

=

(1 + a2 t)(1 + a2 q −2 t)(1 + a2 q 2 t3 ) , (1 − q 2 )(1 − q −4 t−2 )(1 − q 6 t2 )

3

P1 2

= P red 1 2 3

(1 + a2 q −2 t)(1 + a2 q 2 t3 ) , (1 − q 4 t2 )(1 − q −6 t−4 )

P red = 1 3 2

(1 + a2 q −2 t)(1 + a2 q 2 t3 ) . (1 − q −4 t−2 )(1 − q 6 t2 )

For the unreduced sl N -homology we have: P d1N 2 =

(5)

3

P d1N 3 =

(6)

2

(1 + q 2N t)(1 − q 2N −2 − q 2N +2 t2 + q 2N +4 t2 + q 2N +4 t3 − q 4N t3 ) (1 − q 2 )(1 − q 4 t2 )(1 − q −6 t−4 ) (1 + q 2N t)(1 − q 2N −2 − q 2N +2 t2 + q 2N +4 t2 + q 2N +4 t3 − q 4N t3 ) . (1 − q 2 )(1 − q −4 t−2 )(1 − q 6 t2 )

For the reduced sl N -homology we have (1 + q 2N −2 t)(1 + q 2N +2 t3 ) (1 + q 2N −2 t)(1 + q 2N +2 t3 ) dN ,red , P (N > 2); = 1 3 (1 − q 4 t2 )(1 − q −6 t−4 ) (1 − q −4 t−2 )(1 − q 6 t2 )

P d1N ,red = 2 3

2

P d12 ,red = 2 3

2

6 3

(1 − q )(1 + q t ) , (1 − q 4 t2 )(1 − q −6 t−4 )

P d12 ,red = 3 2

(1 + q 2 t) = P d12 ,red . (1 − q −4 t−2 ) 1

These can also be interpreted as homology of dN acting on some algebras. 1 2

For the diagram 3 , the HOMFLY-PT homology is generated by x0 , x1 , b2 , ξ0 , ξ1 , θ1 , and the differential has the form N N −2 2 N N −1 N −1 dN (ξ0 ) = x0 , dN (ξ1 ) = N x0 x1 , dN (θ1 ) = N x0 + x x 1 b2 . 2 0 Note that this differential can be obtained from the symmetric one: we set θ1 = ξ2 /x2 and b2 = 1/x2 , so N N −2 2 1 1 N −1 x x1 ). dN (θ1 ) = dN (ξ2 ) = (N x0 x2 + x2 x2 2 0 1 3

For the diagram 2 , the HOMFLY-PT homology is generated by x0 , a1 , x2 , ξ0 , θ1 , ξ2 and the differential has the form N N −2 N −1 N −1 dN (ξ0 ) = x0 , dN (θ1 ) = N x0 , dN (ξ2 ) = x x2 . 2 0 One can check that the Poincaré series for the homology of dN agree with (5) and (6). Finally, we define d0 for the diagram 1 2 3

1 2 3

by the equation d0 (θ1 ) = x1 b2 , and for the diagram

by the equation d0 (ξ2 ) = x2 a1 . One can check that d0 ,red

P1 3

2

(1 − q −2 t−2 )(1 + a2 q 2 t3 ) = , (1 − q 4 t2 )(1 − q −6 t−4 )

d0 ,red

P1 2

3

(1 − q 2 )(1 + a2 q −2 t) = . (1 − q −4 t−2 )(1 − q 6 t2 )

14


3.8. Homology of (3, m)-torus knots. We summarize the known evidence for the Conjecture 2 in the following theorem. Theorem 15. 1. For the HOMFLY-PT homology, Conjecture 2 agrees with the “refined Chern-Simons invariants” defined in [1]. The conjecture is equivalent to the main conjecture of [11]. 2. The conjecture holds for sl 2 -homology. 3. The reduced homology of d0 agrees with the hat version of the Heegaard-Floer homology of (3, m)-torus links after regrading a = t−1 (cf. [7, Section 3.8]). Proof. The refined Chern-Simons invariants have been been presented as sums over SYT in [9], and the conjectural HOMFLY-PT homology of (3, m)-torus knots was first described in [7, Conjecture 6.8] and later reformulated in [11]. The sl 2 -homology were computed by Turner in [20], and one can check his answers agree with the ones provided by Conjecture 2: P(T (3, 3k + 1), d2 ) =

P(T (3, 3k + 2), d2 ) =

1 + q 2 + q 4 t2 + q 8 t3 + q 10 t5 + q 12 t5 + 1 − q 6 t4 4 (1 − q 6 t2 )(1 + q 4 t) 6k 4k 1 + q t + q t q 6k t4k , (1 − q 4 t2 )(1 − q −6 t−4 ) 1 − q −4 t−2 1 + q 2 + q 4 t2 + q 8 t3 + q 10 t5 + q 12 t5 + 1 − q 6 t4 4 (1 − q 6 t2 )(1 + q 4 t) 6k+4 4k+2 1 + q t q 6k t4k + q t . (1 − q 4 t2 )(1 − q −6 t−4 ) 1 − q −4 t−2

Similar decomposition for sl 2 homology of (3, m)-torus knots was obtained in [15]. The Heegaard-Floer homology of torus knots is well known, see e.g [7, Section 6.12] for its description for (3, m)-torus knots. One can compare it with: P red (T (3, 3k + 1), d0 ) =

P red (T (3, 3k + 2), d0 ) =

−2 −2 2 1 + q2t 6k 4k (1 − q t )(1 + q t) + q t + 1 − q 6 t4 (1 − q 4 t2 )(1 − q −6 t−4 ) −2 −1 (1 − q 2 )(1 + q −2 t−1 ) 12k 6k 1 + q t q 6k t4k + q t , (1 − q −4 t−2 )(1 − q 6 t2 ) 1 − q −6 t−2 −2 −2 2 1 + q2t 6k 4k (1 − q t )(1 + q t) + q t + 1 − q 6 t4 (1 − q 4 t2 )(1 − q −6 t−4 ) −2 −1 (1 − q 2 )(1 + q −2 t−1 ) 12k+4 6k+2 1 + q t q 6k+4 t4k+2 + q t . (1 − q −4 t−2 )(1 − q 6 t2 ) 1 − q −6 t−2

For N = 3, we checked the conjecture up to the (3, 83)-torus knot. 4. DATA FOR STABLE sl 3 - HOMOLOGY In this section, the stable sl 3 -homology of torus knots is compared against FoamHo calculations [14], which were conducted on a Xeon CPU E5-2620 with 128 GB RAM. Note that the quantum degree of the stable part of the sl 3 -homology of an actual (n, m)-torus knot is shifted by 2(nm − n − m) relative to the homology of the (n, ∞)-torus knot.


15

4.1. n = 2. The unreduced Poincaré series is given by (4): P2alg (sl 3 , Q) =

1 − q 6 − q 8 t2 + q 10 t2 + q 10 t3 − q 14 t3 . (1 − q 2 )(1 − q 4 t2 )

The reduced Poincaré series equals: P2alg,red (sl 3 , Q) =

1 + q 8 t3 . 1 − q 4 t2

All of homology is stable, e.g. the first divergence when comparing with the (2, 201)-torus knot occurs at homological degree 202. 4.2. n = 3. The unreduced Poincaré series is given by Theorem 13: P3alg (sl 3 , Q)(1 − q 2 )(1 − q 4 t2 )(1 − q 6 t4 ) = 1−q 6 −q 8 t2 +q 10 t2 +q 10 t3 −q 14 t3 −q 10 t4 +q 12 t4 +q 12 t5 −q 16 t5 −q 18 t7 +q 22 t7 +q 22 t8 −q 24 t8 . The reduced Poincaré series equals: P3alg,red (sl 3 , Q) =

(1 + q 8 t3 )(1 + q 10 t5 ) . (1 − q 4 t2 )(1 − q 6 t4 )

Compared with the rational homology of the (3, 83)-torus knot (calculation took 6 hours and 1.5 GB RAM), the first divergence is at q 168 t112 , both for unreduced and for reduced homology. 4.3. n = 4. The computations of the Koszul homology were done with Singular [6], a computer algebra system. The unreduced Poincaré series equals: P4alg (sl 3 , Q)(1 − q 2 )(1 − q 4 t2 )(1 − q 6 t4 )(1 − q 8 t6 ) = 1−q 6 −q 8 t2 +q 1 t2 +q 10 t3 −q 14 t3 −q 10 t4 +q 12 t4 +q 12 t5 −q 16 t5 −q 12 t6 +q 14 t6 +q 14 t7 −2q 18 t7 + q 22 t7 +q 22 t8 −q 24 t8 −q 20 t9 +q 24 t9 +q 24 t10 −q 26 t10 −q 22 t11 +q 26 t11 +q 26 t12 −q 28 t12 −q 30 t14 +q 36 t14 The reduced Poincaré series equals: P4alg,red (sl 3 , Q) =

(1 + q 8 t3 )(1 + q 10 t5 )(1 − q 12 t6 ) . (1 − q 4 t2 )(1 − q 6 t4 )(1 − q 8 t6 )

Compared with the rational homology of the (4, 35)-torus knot (computation took 12 days and 25 GB RAM), the first divergence is at q 72 t54 , both for unreduced and for reduced homology. 4.4. n = 5. The unreduced Poincaré series (again computed by Singular) equals: P5alg (sl 3 , Q)(1 − q 2 )(1 − q 4 t2 )(1 − q 6 t4 )(1 − q 8 t6 )(1 − q 10 t8 ) = 1−q 6 −q 8 t2 +q 10 t2 +q 10 t3 −q 14 t3 −q 10 t4 +q 12 t4 +q 12 t5 −q 16 t5 −q 12 t6 +q 14 t6 +q 14 t7 −2q 18 t7 + +q 22 t7 −q 14 t8 +q 16 t8 +q 22 t8 −q 24 t8 +q 16 t9 −2q 20 t9 +q 24 t9 +q 24 t10 −q 26 t10 −2q 22 t11 +2q 26 t11 + 2q 26 t12 −2q 28 t12 −q 24 t13 +q 28 t13 +q 26 t14 −2q 30 t14 +q 36 t14 −q 28 t15 +q 30 t15 +q 32 t15 −q 34 t15 +q 30 t16 −q 32 t16 −q 34 t16 +q 38 t16 +q 34 t17 −q 36 t17 −q 36 t18 +2q 40 t18 −q 42 t18 +q 36 t19 −q 38 t19 +q 40 t19 −q 42 t19 − q 38 t20 + 2q 42 t20 − q 44 t20 + q 42 t21 − q 44 t21 + q 44 t22 − q 46 t22 − q 46 t23 + q 50 t23 .

16


The reduced Poincaré series equals: (1 + q 8 t3 )(1 + q 10 t5 )(1 − q 12 t6 − q 14 t8 + q 18 t10 + q 18 t11 − q 26 t15 ) (1 − q 4 t2 )(1 − q 6 t4 )(1 − q 8 t6 )(1 − q 10 t8 ) Compared with the rational homology of the (5, 14)-torus knot (computation took 15 days and 33 GB RAM), the first divergence is at q 30 t24 , both for unreduced and for reduced homology. P5alg,red (sl 3 , Q) =

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